The methodology of design of satellite working mechanism of positive displacement machine

In this paper is described a methodology of design of satellite mechanism consisting of two non-circular gears (externally toothed rotor and internally toothed curvature) and circular gears (satellites). In the presented methodology is assumed that the rotor pitch line is known, and the curvature pitch line is necessary to designate. The presented methodology applies to mechanisms for which the number of the curvature humps is at least one greater than the number of rotor humps. The selection of the number of gears and the number of teeth in gear and rotor and curvature is also presented. The methodology of calculating the position of the satellite center and the angle of its rotation in order to shape the teeth on the rotor and curvature is presented. The article is also showed different types of satellite mechanisms—satellite mechanisms with the different numbers of humps on the rotor and curvature. The technical parameters of the mechanism for the rotor pitch line described by the cosine function are also presented.

www.nature.com/scientificreports/ the available tools (number of chisel teeth, its diameter etc.) should also be taken into account. Moreover, the phenomenon of tooth interference in the mechanism had to be avoided 43 . Therefore, Kujawski in 43 formed the rotor of the type 4 × 6 mechanism by the arcs of circles connected by straight sections and the construction of curvature consists of only arcs of circles. Kujawski was the first to present the guidelines and basics of designing the satellite mechanisms 43 .
In work 26 a four-humped rotor is composed of arcs tangent to each other are presented (Fig. 5). Dowel Li et al. also developed a methodology of designing the satellite mechanism type 4 × 6 based on circular arc curves of the rotor and curvature 44 .
In mechanisms type 3 × 4, 4 × 6 and 6 × 8 the shape of the rotor consists of arcs of circles with different radii and tangents to each other. It was shown in 26 that in these mechanisms are unfavourably large changes in the satellite's acceleration at the moment of transition from the convex part of rotor to the concave part, i.e. in point where arcs of circles meet (Fig. 6). The immediate cause of this is a jump change in the radius value at the point of tangency of the circle's arcs. Thus, in an operating mechanism, especially at high rotational speed, there will be a large mechanical loss which contributes to accelerated wear of the teeth. Figure 7 shows that the wear occurs not only at the points of contact of the arcs but also on the convexity (on the hump) of the rotor. The reason for this is the small radius of this hump 45 .       www.nature.com/scientificreports/ Currently is possible to manufacture the toothed elements using the wire electrical discharge machining (the so-called WEDM method). This method is already used to manufacture well-known satellite mechanisms, especially the 4 × 6 type, the structures of which were designed to be made by chiseling 44,47,48 . Thus, the WEDM makes it possible to manufacture satellite mechanisms with different rotors and curvature shapes. For example, the rotor of the mechanism type 4 × 5 (Fig. 4) has a circular-sinusoidal shape 35 . The radius r R of the rotor pitch line was described by equation 35 : where r Rmin and r Rmax -respectively: minimum and maximum rotor radius, n R -the number of the rotor humps, α R -angle (Fig. 13).
For the mechanism shown in Fig. 4 is: r Rmin = 22,552 mm and r Rmax = 27.524 mm 35 . Nowadays are proposed the next concepts of satellite mechanism. Osiecki proposes a satellite mechanism type 2 × 4 with the elliptical shape of the rotor (two humps) and with four humps curvature 45,46 . However, Osiecki did not disclose the methodology of designing curvature and methodology of selection the number of teeth in the mechanism elements (rotor, curvature and satellite). In literature the satellite mechanisms type 2 × 2 and 2 × 3 are also known (Fig. 8) [49][50][51][52][53][54][55][56] .
Volkov at all propose to design satellite mechanisms, like shown in Fig. 8, by specifying, first, the trajectory of the satellite's center associated with the rotor and with the curvature. In polar coordinates the distance L SR of the satellite center associated with the rotor from the origin of the coordinate system is 49-52 : and the distance L SE of the satellite center associated with the curvature from the origin of the coordinate system is 49-52 : where f(…)-cyclic function, k t -the coefficient that characterizes the non-circularity of the trajectories, L C -the radius of the circle to which both trajectories degenerate when k = 0, α SR and α SE -polar angles associated with the rotor and curvature respectively.    www.nature.com/scientificreports/ Next, the rotor and curvature curves are calculated as equidistance's of the above-mentioned trajectories (2) and (3), assuming the satellite diameter and the number of teeth on the satellite, rotor and curvature [49][50][51][52] .
Zhang et al. 57 and Wang et al. 39 propose the new satellite mechanism type 4 × 6 with the high-order ellipse shape of the rotor. The pitch curve of the rotor is described in polar coordinates as: where r R -the distance between the origin of the coordinate system and a point on the rotor pitch curve, k e -the eccentricity of the ellipse, A-the long-axis radius of the ellipse, α R -the polar angle of the rotor pitch curve.
Zhang at all are also propose to describe the rotor pitch line as 57 : where r c -the radius of basic circle, A h -the amplitude of the cousine function, B-coefficient. According to Zhang, Wang at all the equation describing the curvature pitch curve in polar coordinates is 39,57 : where r E -the distance between the origin of the coordinate system and a point on the curvature pitch curve, r S -radius of the satellite pitch curve, α E -the polar angle of the curvature pitch curve. Zhang at all indicate that with an inapproprioprate selection of parameters, the curvature is characterized by self-interlacing of the pitch line ( Fig. 9) or by undercutting the teeth (Fig. 10).
Furthermore, Zhang at all indicate, when selecting the pitch curve parameters, it is necessary to draw the tooth profile of each gear at the same time to judge its feasibility 57 . It is a certain inconvenience. The next problem to solve is determining how to select the coefficients A h and B in Eq. (5) in solving the undercutting problem of the curvature so that the number of satellite teeth z S is small enough 57 .
In the above-characterized known methods of designing a satellite mechanism, some disadvantages related to the selection of the parameters of the rotor and curvature pitch lines as well as the selection of the number of teeth and their module can also be seen. Therefore, new two methods for designing any type of satellite mechanism (4)   www.nature.com/scientificreports/ for n E > n R are proposed below. The first method allows determining parameters of satellite mechanism for the perfect solution and the second method allows for the correction of the teeth.

Proposed method of designing a satellite mechanism
In the method of designing a satellite mechanism it is assumed that the satellite plays the role of a chisel. That is in a computer program the satellite chisels the gears of the rotor and curvature. Therefore, the mathematical relationships describing the position of the satellite center in the X-Y coordinate system and the corresponding angle of satellite rotation are also presented below.
Basic conditions. It is assumed that for each radius r s of the satellite's pitch circle and for each rotor pitch  11); (g) if the curves of the rotor are between points R 1 and R 2 and the curves of curvature are between points E 1 and E 2 ( Fig. 11) and also the tangents to these curves are perpendicular to the leading radii r R (rotor) and r E (curvature) then the length L Rc of the basic curve of the rotor is equal to the length L Ec of the basic curve of curvature: (h) the central angle β R that covers one half of the cycle of the rotor pitch curve (corresponding to length L Rc ) is: where n R is the number of the rotor humps; (i) the central angle β E that covers one half of the cycle of the curvature's pitch curve (corresponding to length L Ec ) is: Figure 11. The basic geometrical relationships in satellite mechanism. Proposed sequence of proceeding during the design. When starting the design of the satellite mechanism, the first step is to take the number of humps n R on the rotor and the number of humps n E on the curvature. Next the radius r c of the rotor basic circle must be chosen.
When designing a satellite mechanism it should be remembered that: (a) the number of teeth z Rc in the range of rotor angle β R (Fig. 16) is: (b) because the condition (8) must be met, then: (c) the number of teeth z R on the rotor is: (d) the number of teeth z E on the curvature is: (e) numbers of teeth z R and z E must be integer; (f) the number of teeth 2z Rc on the rotor hump (and the same on the curvature hump) must be an integer. Otherwise, despite the fulfilment of the condition (8) and the total number of rotor teeth z R and curvature teeth z E , the satellite mechanism cannot be assembled correctly. An example of such mechanism is presented in Fig. 12.
The first method: searching for the perfect solution. Having the number of humps n R and n E should be sequentially: 1. to adopt radius r c of rotor basic circle; 2. using the iteration method should be searched the amplitude A h and the satellite radius r S until the condition (8) is met with simultaneous fulfilment of the condition (54); 3. using the iteration method should be searched the number of satellite teeth z S so that the numbers of teeth z R and z E are integer; 4. to calculate teeth module m according to the following formula: The calculated value of module m does have to correspond to the normalized value; 5. to adopt the desired normalized value of module m st ; 6. all the parameters of the satellite mechanism should be scaled by a value: Figure 12. Satellite mechanism type 4 × 6 with incorrectly selected parameters: z S = 9, z Rc = 4.75, z R = 38, z E = 57. Every second satellite cannot be inserted into the mechanism. The search for the parameters of satellite mechanism, ie A h , r S and z S so that the condition (54) was met may prove impossible. Then it is worth to allowing a certain difference δ of the lengths L Rc and L Ec , but not greater than the limit value δ b , that is: The value δ b is justified, for example, by the fact that depending on the processing technology, different accuracy of the manufacturing of satellite mechanism elements is obtained.
The second method. Having the number of humps n R and n E should be sequentially: 1. to adopt the basic circle radius r c of the rotor; 2. to adopt the amplitude A h ; 3. by iteration method to search the satellite radius r S until the condition (8) is met while meeting the condition (54) (presented in "Method 2"); 4. to adopt the number of teeth z Rc (keeping in mind that the number of teeth 2z Rc on the rotor hump should be integer); 5. to calculate module m by transforming the formula (11); 6. to calculate the number of teeth z S by transforming the formula (15). The obtained value z S need not be an integer; 7. if the calculated value of z S is not integer then the total number of the satellite teeth z Sst should be adopted.
It is recommended that: 8. to apply the P-O correction of the teeth. The minimum value of the correction coefficient is: 9. in order to generate corrected teeth of the rotor and the curvature, the angle γ Sst of satellite rotation relative to its center S should be calculated according to the following formula: where: Rotor designing: the coordinates of the rotor pitch line. According to the basic conditions shown in "Basic conditions" the rotor's pitch lines should be a cyclically changing curve. Furthermore, the rotor should have a total number of humps n R evenly distributed over the entire circumference of the rotor. Therefore, the radius r R of the rotor pitch line can be defined by any cyclical function type r R = f(α R ), for example functions (1), (4), (5) and any others. A schematic sketch of a quarter rotor with the basic geometrical dimensions is shown in Fig. 13. The coordinates (x R ,y R ) of the point R on the rotor pitch line are: The most distant point of the rotor pitch line from the center of rotation of this rotor (i.e. from the origin of the coordinate system) designates the straight-line k which is the axis of symmetry of the rotor hump (Fig. 13). If α S = α R = β R then the satellite center S and the tangent point R of the satellite with the rotor lie on the straightline k (Fig. 16b). The angle β R can be calculated from the formula (10).
The coordinates of the satellite center. The satellite pitch circle with radius r s must be tangent in the point R to the rotor pitch line (Fig. 14).
The coordinates (x S ,y S ) of the satellite center S can be calculated according to the following formulas: Vol.:(0123456789) where a (R)p is the slope of the straight-line y (R)p perpendicular to the tangent line y (R)t in the point R (Fig. 14). If a (R)p < 0 then in formulas (25) and (26) is a "−" sign instead of the "±" sign. But for a (R)p ≥ 0 is sign "+". The angular position of the satellite center S (angle α S in Fig. 14) about the axis OY can be calculated from the following formula: whereas the distance L S of satellite center S from the origin of the coordinate system is: The angle of satellite rotation. Each position of satellite center S described by formulas (25) and (26) is assigned the angle γ S of the satellite rotation around the center S (Fig. 15).
The angle γ S of satellite rotation around its center S can be calculated according to the following formula:   (Fig. 16). Furthermore for r R = r Rmin is r E = r Emin and for r R = r Rmax is r E = r Emax . The number n E of curvature humps corresponds to the angle β E , which can be calculated from the formula (10). Formulas (9) and (10) follow that: and hence: Furthermore, if the satellite moves in relation to the rotor by the angle β R then the curvature will rotate by the angle β Ec (Fig. 16):    (31) and (32) exist between the angles β R , β E and β Ec then the following relations between the angles α S , α SE and θ E are also true (Fig. 17): In order to determine the curvature pitch line, the set of coordinates (x E ,y E ) of the point E should be calculated. These coordinates can be calculated by two methods.

Method 1: according to three temporaries centers of rotation.
For any position of the satellite the tangency point E of the satellite with the curvature lies on the straight line that passes through the center of rotation of the rotor and the tangency point R of the satellite with the rotor (Fig. 18).
The coordinates of the point E are as follows:  For any location of the satellite center S relative to the rotor corresponds the location of the center S E of this satellite relative to the curvature (Fig. 19). The satellite pitch circle with center in point S E is tangent the pitch line of the curvature. The angle ρ between points S and S E can be calculated from the following formula: whereas the coordinates (x SE ,y SE ) of the point S E are: where L S is expressed by the formula (28).
The coordinates (x E ,y E ) of the point E (Fig. 19) can be calculated from formulas: where:  (1) of the satellite with the curvature rotate with the angle ∆α E (1) and is in a new position E (1) (Fig. 20). The satellite pitch circle in a new position (point S (2) according to Fig. 20) is tangent to the curvature pitch line. The satellite center is common for satellite position relative to the rotor and satellite position relative to the curvature, that is S (2) = S E (2) . The next displacement of the satellite with the angle ∆α S (2) (Fig. 21) forces the rotation of both the satellite center S (2) and the point E (1) with the angle ∆α E (2) . Therefore the new position of satellites relative to the curvature are S E (1) , S E (2) and S E (3) . Wherein, the satellite center position S E (3) is the same as the position S (2) relative to the rotor, that is S (3) = S E (3) . In the next steps the same is done until α S = α R = β R . The displacement of the satellite with the angle ∆α S (1) causes the tangency point R will travel the distance ∆L R (1) (Fig. 20). The same distance must travel the tangency point E, that is:   (Fig. 16). The next displacement of the satellite with the angle ∆α S (2) forces the rotation of the points E (1) and E (2) (i.e. the current point and the previous points) with the angle ∆α E (2) . In the next steps, the same is done until α S = α R = β R .
It is more advantageous to determine the curvature pitch line only after determining the entire set of satellite centers S E , i.e. after determining all centers S E with the step of ∆α E ). This method is illustrated in Fig. 22. The circles with the radius r E and with the center C E are tangent to the three next satellites. That is, the circle with the radius r E (2) is tangent to satellites 1, 2 and 3. The point E (2) is the tangency point of the circle with the radius r E (2) with the satellite 2. Similarly, the circle with the radius r E (3) is tangent to the satellites 2, 3 and 4 and also the point E (3) is the tangency point of the circle with the radius r E (3) with the satellite 3.
Analytical method of determinig curvature pitch line. The coordinates (x E (1) ,y E (1) ) of the first point E (1) of the curvature (Fig. 22) should be calculated from formulas (43) and (44). But coordinates (x E (i) ,y E (i) ) of the next points E (i) of the curvature should be calculated according to the method shown in Fig. 22. That is, the coordinates (x CE (i) ,y CE (i) ) of the center of the circle with the radius r E (i) (with the number (i)) can be calculated from formulas: But the coordinates (x E (i) ,y E (i) ) of the curvature point E (i) are: where: Then there is no self-interlacing of the curvature pitch line like in Fig. 9. The mathematical formulas presented above allow calculating the coordinates of the point E on the curvature pitch line only in the range of the angle β E . The points in the second half of the curvature hump are a mirror image of the points E with respect to the straight-line k. But the total pitch line of the curvature is the circular array of the hump pitch line with respect to the origin of the coordinate system.

The length of the rotor pitch line and the curvature pitch line
The length L R of the rotor pitch line in terms of the angle α R is the sum of the elementary lengths ∆L R (i) defined by two adjacent points (R (i) and R (i+1) ) of the rotor pitch line (Fig. 23), that is: Similarly, the length L E of the curvature pitch line in the range of angle α E is the sum of the elementary lengths ∆L E (i) defined by two adjacent points (E (i) and E (i+1) ) of the curvature pitch line (Figs. 19,22,23), that is: If α R = β R then L R = L Rc and if α E = β E then L E = L Ec .

The angle between satellites and the number of satellites
If the angle β E corresponds to curvature hump then the rotate of the curvature by the angle will enable the placement of the next satellite in the same place (point S (1) -Fig. 24). The formulas (10) and (34) indicate that the angle φ S between satellites is: The number n S of satellite in mechanism is:

Types of satellite mechanisms
By the type of satellite mechanism should be understood its characteristic future, i.e. the number n R of the rotor humps and the number n E of the curvature humps. Therefore the type of mechanism is marked as "n R x n E ". If the number n E of the curvature humps increases in relation to the number n R of the rotor humps then increases the satellite diameter and decreases the distance between axes of two adjacent satellites. Thus, in a satellite mechanism of any type, each pair of adjacent satellites must satisfy the following condition: -coordinates of two adjacent satellites, h hs -the tooth head height. The Types of satellite mechanisms that meet the condition (60) are shown in Fig. 25. It should be noted that, regardless of the type of mechanism, the distance between the satellite's pitch diameters decreases if the difference in the number of rotor and curvature humps increases. Therefore these mechanisms are characterized by a large number of teeth and large dimensions.
Is not possible to build a satellite mechanism if: The parameters of satellite mechanism for the cosinusoidal shape of the rotor pitch line Below were determined the satellite mechanism parameters for the radius r R of the rotor pitch curve expressed as ( Fig. 13): where: That is a cosine curve with an amplitude A h is "wound" on the circle with a radius r c . The coordinates (x R ,y R ) of the point R on the rotor pitch line are: The slope a (R)p f the straight-line y (R)p perpendicular to the tangent line y (R)t in the point R is: Table 1 summarizes the parameters of selected satellite mechanisms calculated using the first method (see "The first method: searching for the perfect solution"). Whereas Table 2 summarizes parameters of selected satellite mechanisms calculated using the second method (see "The second method"), assuming:  In both cases the parameters were determined for tooth module m = 1 mm. Development of satellite mechanisms with n R > 8 is possible but these mechanisms are unlikely to find technical application.

Verification of satellite mechanism designing method
The parameters of satellite mechanism type 4 × 5, shown in Table 1, after scaling to the module m = 0,5 mm correspond to the parameters of the mechanism shown in Fig. 4. Thus, it confirms the correctness of the design methodology and the performed calculations. In order to verify the presented procedure of designing, a sample satellite mechanism was designed, manufactured and examined. The projects of rotor and curvature, created according to the above-described methodology are presented in Figs. 26 and 27. But Fig. 28 shows satellite mechanism type 4 × 6 made of metal by WEDM.
The satellite mechanism was checked for smooth rotation.

Summary
This article presents the method of designing a satellite mechanism based on the adopted function of the rotor pitch line. The sequence of the proceedings for selecting the parameters of the satellite mechanism is described. All technically possible types of satellite mechanisms are also presented. These mechanisms were calculated assuming the sinusoidal function to determine the shape of the rotor pitch line. From the point of view of the construction of hydraulic displacement machines, the use of a rotor with a circular-sinusoidal shape is justified. www.nature.com/scientificreports/ Both the shape of the rotor teeth and the curvature teeth are determined by the shape of the satellite teeth. The satellite is a gear-shaper cutter. In this way the teeth interference was eliminated. This is an undoubted advantage of the presented method. Furthermore, the presented method of design allows to avoid the self-intersection of curvature pitch line and undercutting the curvature teeth. Is possible to apply a teeth correction and to create a satellite mechanism even with a very small number of satellite teeth (for example z S = 8).
The practical verification shown that using the presented method of design has good results. In the manufactured mechanism, no problems with gears meshing were observed-the mechanism worked smoothly, without jamming.   The results presented in this paper can constitute fundamentals for further investigations of other particular properties of different types of satellite mechanisms, such as: • the volume of the working chamber as a function of the angle of rotation of the rotor or curvature. The working chamber should be understood as a volume formed by two adjacent satellites, rotor and curvature; • geometrical displacement of a hydraulic machine with different types of satellite mechanisms; • theoretical characteristics of torque and flow rate in a mechanism and thus an irregularity of flow rate and torque also; • satellite kinematics and dynamics; • mechanical losses.
Undoubtedly, an important issue will be the analysis of the impact of tooth structure (e.g. involute teeth, circular-arc teeth, etc.) on their strength.
Nevertheless, the methodology of designing the curvature is universal for different shapes of the rotor. The methodology of designing the satellite mechanism, presented below, enables their manufacturing using the wire electrical discharge machining method.

Data availability
The datasets generated and/or analysed during the current study are not publicly available due to patent protection of the solutions presented in the article (patent applications P.437749, P.437750 and P.437751) but are available from the corresponding author on reasonable request.